We prove that, under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines that have disconnected realization spaces.

We show that the γ-vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In particular, we prove that such γ-vector is the f -vector of some balanced simplicial complex. Moreover, we show that the local γ-vector of the interval subdivision of a simplex is nonnegative; answering a question by Juhnke-Kubitzke et al.Conjecture 1.1 is a strengthening of the well known Charney-Devis conjecture. The Gal conjecture holds for all Coxeter complexes (see [Ste08]), for the dual simplicial complexes of associahedron and cyclohedron (see [NP11]), and for barycentric subdivision of homology sphere (see[NPT11]). The authors in [NP11] conjectured further strengthening of Gal conjecture. Conjecture 1.2. [NP11, Problem 6.4] If ∆ is a flag homology sphere then γ(∆) is the f -vector of some balanced simplicial complex.This conjecture holds for the dual simplicial complex of all flag nestohedera, see in [Ais14]. Frohmader [Fro08, Theorem 1.1] showed that the f -vector of any flag simplicial complex satisfies the Frankl-Füredi-Kalai (FFK) inequalities (see [FFK88]). In [NPT11], authors showed that the γ-vector of the barycentric subdivision of a homology sphere satisfies the FFK inequalities, i.e., the f -vector of a balanced simplicial complex. The first aim of this paper is the confirmation of Conjecture 1.2 in the case of the interval subdivision of a homology sphere. The main theorem is stated as: Theorem 1.3. If ∆ is a simplicial complex with a nonnegative and symmetric h-vector, then the γ-vector of the interval subdivision of ∆ is the f -vector of a balanced simplicial complex.

Nekrasov-Okounkov identity gives a product representation of the sum over partitions of a certain function of partition hook length. In this paper we give several generalizations of the Nekrasov-Okounkov identity using the cyclic symmetry of the topological vertex.

A rank one local system L on a smooth complex algebraic variety M is admissible roughly speaking if the dimension of the cohomology groups H m (M, L) can be computed directly from the cohomology algebra H * (M, C).We say that a line arrangement A is of type C k if k ≥ 0 is the minimal number of lines in A containing all the points of multiplicity at least 3. We show that if A is a line arrangement in the classes C k for k ≤ 2, then any rank one local system L on the line arrangement complement M is admissible. Partial results are obtained for the class C 3 .

In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex ∆ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in [8]). Moreover, we introduce the concept of Gallai-simplicial complex ∆ Γ (G) of a finite simple graph G. As an application, we show that the face ring of the Gallai simplicial complex associated to tree is Cohen-Macaulay.